thomas villmann, erzsébet merényi and barbara hammer- neural maps in remote sensing image analysis

8/3/2019 Thomas Villmann, Erzsbet Mernyi and Barbara Hammer- Neural Maps in Remote Sensing Image Analysis

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Neural Maps in Remote Sensing Image Analysis

Thomas Villmann, Erzsbet Mernyi and Barbara Hammer

Universitt Leipzig, Germany

Klinik fr Psychotherapie, 04107 Leipzig, K.-Tauchnitz-Str.25

Rice University

Depatment of Electrical and Computer Engineering

6100 Main Street, MS 380

Houston, Texas, U.S.A.

University of Osnabrck,

Department of Mathematics/Computer Science

Albrechtstrae 28, 49069 Osnabrck, Germany

corresponding author; email: [email protected], phone: +49 (0) 341 9718868 fax: +49

(0) 341 2131257


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Abstract

We study the application of Self-Organizing Maps for the analyses of remote sensing

spectral images. Advanced airborne and satellite-based imaging spectrometers produce very

high-dimensional spectral signatures that provide key information to many scientific inves-

tigations about the surface and atmosphere of Earth and other planets. These new, so-

phisticated data demand new and advanced approaches to cluster detection, visualization,

and supervised classification. In this article we concentrate on the issue of faithful topo-

logical mapping in order to avoid false interpretations of cluster maps created by an SOM.

We describe several new extensions of the standard SOM, developed in the past few years:

the Growing Self-Organizing Map, magnifi

cation control, and Generalized Relevance Learn-ing Vector Quantization, and demonstrate their effect on both low-dimensional traditional

multi-spectral imagery and 200-dimensional hyperspectral imagery.


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1 Introduction

In geophysics, geology, astronomy, as well as in many environmental applications airborne

and satellite-borne remote sensing spectral imaging has become one of the most advanced

tools for collecting vital information about the surface of Earth and other planets. Spec-

tral images consist of an array of multi-dimensional vectors each of which is assigned to a

particular spatial area, reflecting the response of a spectral sensor for that area at various

wavelengths (Figure 1). Classification of intricate, high-dimensional spectral signatures has

turned out far from trivial. Discrimination among many surface cover classes, discovery

of spatially small, interesting spectral species proved to be an insurmountable challenge to

many traditional clustering and classification methods. By costumary measures (such as,

for example, Principal Component Analysis (PCA)) the intrinsic spectral dimensionality of

hyperspectral images appears to be surprisingly low, 5 10 at most. Yet dimensionality

reduction to such low numbers has not been successful in terms of preservation of impor-

tant class distinctions. The spectral bands, many of which are highly correlated, may lie on

a low-dimensional but non-linear manifold, which is a scenario that eludes many classical

approaches. In addition, these data comprise enormous volumes and are frequently noisy.

This motivates research into advanced and novel approaches for remote sensing image analy-sis and, in particular, neural networks [1]. Generally, artificial neural networks attempt to

replicate the computational power of biological neural networks, with the following important

(incomplete list of) features:

adaptivity the ability to change the internal representation (connection weights, net-

work structure) if new data or information are available

robustness handling of missing, noisy ore otherwise confused data

power / speed handling of large data volumes in acceptable time due to inherent

parallelism

non-linearity the ability to represent non-linear functions or mappings

Exactly these properties make the application of neural networks interesting in remote

sensing image analysis. In the present contribution we will concentrate on a special neural

network type, neural maps. Neural maps constitute an important neural network paradigm.


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In brains, neural maps occur in all sensory modalities as well as in motor areas. In tech-

nical contexts, neural maps are utilized in the fashion of neighborhood preserving vector

quantizers. In both cases these networks project data from some possibly high-dimensional

input space onto a position in some output space. In the area of spectral image analysis we

apply neural maps for clustering, data dimensionality reduction, learning of relevant data

dimensions and visualization. For this purpose we highlight several useful extensions of the

basic neural map approaches.

This paper is structured as follows: In Section 2 we briefly introduce the problems in

remote sensing spectral image analysis. In Section 3 we give the basics of neural maps and

vector quantization together with a short review of new extensions: structure adaptation

and magnification control (Section 3.2), and learning of proper metrics or relevance learning

(Section 3.3). In Section 4 we present applications of the reviewed methods to multi-spectral

LANDSAT TM imagery as well as very high-dimensional hyperspectral AVIRIS imagery.

2 Remote Sensing Spectral Imaging

As mentioned above airborne and satellite-borne remote sensing spectral images consist of

an array of multi-dimensional vectors assigned to particular spatial regions (pixel locations)

reflecting the response of a spectral sensor at various wavelengths. These vectors are called

spectra. A spectrum is a characteristic pattern that provides a clue to the surface material

within the respective surface element. The utilization of these spectra includes areas such

as mineral exploration, land use, forestry, ecosystem management, assessment of natural

hazards, water resources, environmental contamination, biomass and productivity; and many

other activities of economic significance, as well as prime scientific pursuits such as looking

for possible sources of past or present life on other planets. The number of applications

has dramatically increased in the past ten years with the advent of imaging spectrometers

such as AVIRIS of NASA/JPL, that greatly surpass traditional multi-spectral sensors (e.g.,

LANDSAT Thematic Mapper (TM)).

Imaging spectrometers can resolve the known, unique, discriminating spectral features

of minerals, soils, rocks, and vegetation. While a multi-spectral sensor samples a given

wavelength window (typically the 0.42.5 m range in the case of Visible and Near-Infrared

imaging) with several broad bandpasses, leaving large gaps between the bands, imaging


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spectrometers sample a spectral window contiguously with very narrow, 1020nm badpasses

(Figure 1) [2]. Hyperspectral technology is in great demand because direct identification of

surface compounds is possible without prior field work, for materials with known spectral

signatures. Depending on the wavelength resolution and the width of the wavelength window

used by a particular sensor, the dimensionality of the spectra can be as low as 5 6 (such

as in LANDSAT TM), or as high as several hundred for hyperspectral imagers [3].

Spectral images can formally be described as a matrix S = v(x,y), where v(x,y) RDV

is the vector of spectral information associated with pixel location (x, y). The elements

v(x,y)i , i = 1 . . . DV of spectrum v

(x,y) reflect the responses of a spectral sensor at a suite

of wavelengths (Figure 2, [4],[5]). The spectrum is a characteristic fingerprint pattern that

identifies the surface material within the area defined by pixel (x, y). The individual 2-

dimensional image Si = vi(x,y) at wavelength i is called the ith image band. The data space

V spanned by Visible-Near Infrared reflectance spectra is [0 noise, U + noise]DV RDV

where U > 0 represents an upper limit of the measured scaled reflectivity and noise is the

maximum value of noise across all spectral channels and image pixels. The data densityP(V)

may vary strongly within this space. Sections of the data space can be very densely populated

while other parts may be extremely sparse, depending on the materials in the scene and on

the spectral bandpasses of the sensor. According to this model traditional multi-spectral

imagery has a low DV value while DV can be several hundred for hyperspectral images. The

latter case is of particular interest because the great spectral detail, complexity, and very

large data volume pose new challenges in clustering, cluster visualization, and classification

of images with such high spectral dimensionality [6].

In addition to dimensionality and volume, other factors, specific to remote sensing, can

make the analyses of hyperspectral images even harder. For example, given the richness

of data, the goal is to separate many cover classes, however, surface materials that are

significantly different for an application may be distinguished by very subtle differences

in their spectral patterns. The pixels can be mixed, which means that several different

materials may contribute to the spectral signature associated with one pixel. Training data

may be scarce for some classes, and classes may be represented very unevenly. All the above

difficulties motivate research into advanced and novel approaches [1].

Noise is far less problematic than the intricacy of the spectral patterns, because of the

high Signal-to-Noise Ratios (500 1, 500) that present-day hyperspectral imagers provide.


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For this discussion, we will omit noise issues, and additional effects such as atmospheric

distortions, illumination geometry and albedo variations in the scene, because these can be

addressed through well-established procedures prior to clustering or classification.

3 Neural Maps and Vector Quantization

3.1 Basic Concepts and Notations

Neural maps as tools for (unsupervised) topographic vector quantization algorithms map

data vectors v sampled from a data manifold V RDV onto a set A of neurons r:

VA : V A . (3.1)

Associated with each neuron is a pointer (weight vector) wr RDV specifying the config-

uration of the neural map W = {wr}rA. The task of vector quantization is realized by the

map as a winner-take-all rule, i.e. a stimulus vector v V is mapped onto that neuron

s A the pointer ws of which is closest to the presented stimulus vector v,

VA : v 7 s (v) = argminrA

d (v,wr) . (3.2)

where d (v,wr) is the Euclidean distance. The neuron s is referred to as the winner or

bestmatching unit. The reverse mapping is defined as AV : r 7 wr. These two mappings

together determine the neural map

M = (VA,AV) (3.3)

realized by the network. The subset of the input space

r = {v V : r = VA (v)} (3.4)

which is mapped to a particular neuron r according to (3.2), forms the (masked) receptive

field of that neuron. If the intersection of two masked receptive fields r, r0 is non-vanishing

we call r and r0 neighbored. The neighborhood relations form a corresponding graph

structure GV in A: two neurons are connected in GV if and only if their masked receptive

fields are neighbored. The graph GV is called the induced Delaunay-graph (See, for example,

[7] for detailed definitions). Due to the bijective relation between neurons and weight vectors,

GV also represents the Delaunay graph of the weights [7].


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In the most widely used neural map algorithm, the self-organizing map (SOM) [8], the

neurons are arranged a priori on a fixed grid A.1 Other algorithms such as the topology

representing network (TRN), which is based on the neural gas algorithm (NG) [9], do not

specify the topological relations among neurons, but construct a neighborhood structure in

the output space during learning [7].

In any of these algorithms the pointers are adapted such that the input data are matched

appropriately. For this purpose a random sequence of data points v V from the stimuli

distribution P(v) is presented to the map, the winning neuron s is determined according to

(3.2), and the pointer ws

is shifted toward v. Interactions among the neurons are included

via a neighborhood function determining to what extent the units that are close to the

winner in the output space participate in the learning step:

4wr = h (r, s,v) (vwr) . (3.5)

h (r, s,v) is usually chosen to be of Gaussian shape

h (r, s,v) = exp

(dA (r, s (v)))2

!(3.6)

where dA (r, s (v)) is a distance measure on the set A. We emphasize that h (r, s,v) im-

plecitely depends on the whole set W of weight vectors via the winner determination of

s according to (3.2). In SOMs it is evaluated in the output space A: dSOMA (r, s (v)) =

kr s (v)kA whereas for the NG we have dNGA (r, s (v)) = kr (v,W). kr (v,W) gives the

number of pointers wr0 for which the relation d (v,wr0) d (v,wr) is valid [9], i.e., dNGA is

evaluated in the input space. (Here, d(, ) is the Euclidean distance.) In the SOM usually

= 22 is used.

The particular definitions of the distance measures in the two models cause further dif-

ferences. In [9] the convergence rate of the neural gas network was shown to be faster than

that of the SOM. Furthermore, it was proven that the adaptation rule for the weight vectors

follows, on average, a potential dynamic. In contrast, a global energy function does not exist

in the SOM algorithm [10].

One important extension of the basic concepts of neural maps concerns the so-called mag-

1 Usually the grid A is chosen as a dAdimensional hypercube and then one has i = (i1, . . . , idA). Yet,

other ordered arrangements are also admissible.


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nification. The standard SOM distributes the pointers according to the input distribution

P(W) P(V) (3.7)

with the magnification factor SOM =2

3 [11], [12].23

For the NG one finds, by analyticalconsiderations, that for small the magnification factor NG =

dd+2

which only depends

on the dimensionality of the input space embedded in RDV, i.e. the result is valid for all

dimensions [9].

Topology preservation or topographic mapping in neural maps is defined as the preser-

vation of the continuity of the mapping from the input space onto the output space, more

precisely it is equivalent to the continuity ofM between the topological spaces with properly

chosen metric in both A and V. For lack of space we refer to [14] for detailed considerations.

For the TRN the metric in A is defined according to the given data structure whereas in case

of the SOM violations of topology preservation may occur if the structure of the output space

does not match the topological structure of V. In SOMs the topology preserving property

can be used for immediate evaluations of the resulting map, for instance by interpretation

as a color space, as demonstrated in Section 4.2.2 . Topology preservation also allows the

applications of interpolating schemes such as the parametrized SOM (PSOM) [15] or inter-

polating SOM (I-SOM) [16]. A higher degree of topology preservation, in general, improves

the accuracy of the map [17].

As pointed out in [14] violations of topographic mapping in SOMs can result in false

interpretations. Several approaches were developed to judge the degree of topology preser-

vation for a given map. Here we briefly describe a variant P of the well known topographic

product P [17]. Instead of the Euclidean distances between the weight vectors, P uses the

respective distances dGV(wr,w

r0) of minimal path lengths in the induced Delaunay-graph GV

of the wr. During the computation of P for each node r the sequences nAj (r) ofj-th neigh-

bors of r in A, and nVj (r) describing the j-th neighbor ofwr in V, have to be determined.

These sequences and further averaging over neighborhood orders j and nodes r finally lead

to

P =1

N(N 1)

Xr

N1Xj=1

1

2jlog() (3.8)

2 This result is valid for the one-dimensional case and higher-dimensional ones which separate.3 The notation magnification factor is, strictly speaking, not correct. Some times it is called (mathemat-

ically exact) magnification exponent. However, the notation magnification factor is commonly used [13].

Therefore we use this notation here.


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with

= jl=1

dGVwr,wnA

l(r)

dGV

wr,wnV

l(r)

dAr,nAl (r)

dA (r,n

Vl (r))

, (3.9)

and dA (r, r0) is the distance between the neuron positions r and r0 in the lattice A. P

can take on positive or negative values with similar meaning as for the original topographic

product P: if P < 0 holds the output space is too low-dimensional, and for P > 0 the

output space is too high-dimensional. In both cases neighborhood relations are violated.

Only for P 0 does the output space approximately match the topology of the input

data. The present variant P overcomes the problem of strongly curved maps which may be

judged neighborhood violating by the original P even though the shape of the map might

be perfectly justified [14].

3.2 Magnification Control and Structure Adaptation

3.2.1 Magnification Control

The first approach to influence the magnification of a vector quantizer, proposed in [18],

is called the mechanism of conscience yielding approximately the same winning probability

for each neuron. Hence, the approach yields density matching between input and output

space which corresponds to a magnification factor of 1. Bauer et al. in [19] suggested a

more general scheme for the SOM to achieve an arbitrary magnification. They introduced

a local learning parameter s for each neuron r with hsi P(wr)m in (3.5), where m is an

additional control parameter. Equation (3.5) now reads as

4wr

= sh (r, s,v) (vwr) . (3.10)

Note that the learning factor s of the winning neuron s is applied to all updates. This local

learning leads to a similar relation as in (3.7):

P(W) P(V)0

(3.11)

with 0 = (m + 1) and allows a magnification control through the choice of m. In partic-

ular, one can achieve a resolution of 0 = 1, which maximizes mutual information [20, 21].

Application of this approach to the derivation of the dynamic of TRN yields exactly the

same result [22].


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3.2.2 Structure Adaptation

For both neural map types, the SOM and the TRN, growing variants for structure adaptation

are known.

In the growing SOM (GSOM) approach the output A is a hypercube in which both the

dimension and the respective length ratios are adapted during the learning procedure in

addition to the weight vector learning, i.e. the overall dimensionality and the dimensions

along the individual directions change in the hypercube. The GSOM starts from an initial

2-neuron chain, learns according the regular SOM-algorithm, adds neurons to the output

space according to a certain criterion to be described below, learns again, adds again, etc.,

until a prespecified maximum number Nmax of neurons is distributed. During this procedure,

the output space topology remains of the form n1 n2 ..., with nj = 1 for j > DA, where

DA is the current dimensionality ofA.4 From there it can grow either by adding nodes in

one of the directions which are already spanned by the output space or by initialising a new

dimension. This decision is made on the basis of the receptive fields r. When reconstructing

v V from neuron r, an error = vwr remains decomposed along the different directions,

which results from projecting back the output grid A into the input space V:

= vwr =

DAXi=1

ai(v) wr+ei wrei

kwr+ei wreik+ v0 (3.12)

Here, ei denotes the unit vector in direction i ofA, wr+ei and wrei are the weight vectors

of the neighbors of the neuron r in the ith direction of the lattice A.5 Considering a re-

ceptive field r and determining the first principle component PCA ofr allows a further

decomposition ofv0. Projection ofv0 onto the direction ofPCA then yields aDA+1(v),

v0

=aDA+1(

v)

PCA

||PCA|| +v00.

(3.13)

The criterion for the growing now is to add nodes in that direction which has, on average,

the largest of the expected (normalized) error amplitudes ai:

ai =

rni

ni + 1

Xvr

| ai(v) |qPDA+1j=1 a

2j(v)

, i = 1,...,DA + 1 (3.14)

4 Hence, the initial configuration is 2 1 1 ..., DA = 1.5 At the border of the output space grid, where not two, but just one neighboring neuron is available, we

usewrwre

i||wrwrei || orwr+e

i

wr

||wreiwr|| to compute the backprojection of the output space direction ei into the input

space.


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After each growth step, a new learning phase has to take place, in order to readjust the map.

For a detailed study of the algorithm we refer to [23].

The Growing TRN adapts the number of neurons in TRN whereby the structure between

them is re-arranged according to the TRN definition [24] Hence, it is capable of representing

the data space in a topologically faithful manner with increasing accuracy and it also realizes

a structure adaptation.

3.3 Relevance Learning

As mentioned above neural maps are unsupervised learning algorithms. In case of labeled

data one can apply supervised classification schemes for vector quantization (VQ). Assuming

that the data are labeled, an automatic clustering can be learned via attaching maps to

the SOM or adding a supervised component to VQ to achieve learning vector quantization

(LVQ) [25, 26]. Various modifications of LVQ exist which ensure faster convergence, a better

adaptation of the receptive fields to optimum Bayesian decision, or an adaptation for complex

data structures, to mention just a few [25, 27, 28].

A common feature of unsupervised algorithms and LVQ is the information provided by

the distance structure in V, which is determined by the chosen metric. Learning heavily

relies on this feature and therefore crucially depends on how appropriate the chosen metric

is for the respective learning task. Several methods have been proposed which adapt the

metric during training [29],[30],[31].

Here we focus on metric adaptation for LVQ since the LVQ combines the elegance of

simple and intuitive updates in unsupervised algorithms with the accuracy of supervised

methods. Let cv L be the label of input v, L a set of labels (classes) with #L = NL. LVQ

uses afi

xed number of codebook vectors (weight vectors) for each class. LetW = {wr} be theset of all codebook vectors and cr be the class label ofwr. Furthermore, let Wc= {wr|cr = c}

be the subset of weights assigned to class c L.

The training algorithm adapts the codebook vectors such that for each class c L, the

corresponding codebook vectors Wc represent the class as accurately as possible. This means

that the set of points in any given class Vc = {v V|cv = c}, and the union Uc =S

r|wrWc

r

of receptive fields of the corresponding codebook vectors should differ as little as possible.

For a given data point v with class label cv let (v) denote some function which is negativeifv is classified correctly, i.e., it belongs to a receptive field r with cr = cv, and which is


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positive ifv is classified incorrectly, i.e., it belongs to a receptive field r with cr 6= cv. Let

f : R R be some monotonically increasing function. The general scheme of GLVQ aims

to minimize the cost function

Cost =Xv

f((v)) (3.15)

via stochastic gradient descent. Different choices of f and (v) yield LVQ or LVQ2.1 as

introduced in [25], respectively, as shown in [27]. Denote by d the squared Euclidean distance

ofv to the nearest codebook vector. The choice of f as the sigmoidal function and

(v) =dr+ d

r

dr+

+ dr

, (3.16)

where dr+

is the squared Euclidean distance of the input vector v to the nearest codebook

vector labeled with cr+ = cv, say wr+, and dr is the squared Euclidean distance to the best

matching codebook vector but labeled with cr 6= cr+, say wr, yields a powerful and noise

tolerant behavior since it combines adaptation near the optimum Bayesian borders similarly

as in LVQ2.1, with prohibiting the possible divergence of LVQ2.1 (as reported in [27]). We

refer to this modification as the GLVQ. The respective learning dynamic is obtained by

differentiation of the cost function (3.15) [27]:

4wr+ =

4 f0|(v) dr

(dr+ + dr)2 (vw

r+) and 4w

r =

4 f0|(v) dr+

(dr+ + dr)2 (vw

r) (3.17)

To result a metric adaptation we now introduce input weights = (1, . . . ,DV), i 0,

kk = 1 in order to allow a different scaling of the input dimensions: Substituting the

squared Euclidean metric by its scaled variant

d (x,y) =

DVXi=1

i(xi yi)2 , (3.18)

the receptive field of codebook vector wr becomes

r=v V : r = VA (v)

(3.19)

with

VA : v 7 s (v) = argminrA

d (v,wr) (3.20)

as mapping rule. Replacing r by

rand d by d in the cost function Cost in (3.15) yields a

variable weighting of the input dimensions and hence an adaptive metric. Now (3.17) reads

as

4wr+ = 4 f0|(v) d

r

(dr+

+ dr

)2(vwr+) and 4wr =

4 f0|(v) d

r+

(dr+

+ dr

)2(vwr) (3.21)


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with being the diagonal matrix with entries 1, . . . ,DV. Appropriate weighting factors

can also be determined automatically via a stochastic gradient descent. In this way the

GLVQ-learning rule (3.17) is augmented as follows:

4k = 1 2 f0|(v)

dr

(dr+

+ dr

)2(v wr+)

2k

dr+(d

r++ d

r)2

(v wr)2k

!(3.22)

with 1 (0, 1) and k = 1 . . . DV, followed by normalization to obtain kk = 1. We term

this generalization of the Relevance Learning Vector Quantization (RLVQ) [32] and GLVQ

Generalized Relevance Learning Vector Quantization or GRLVQ as introduced in [33],[34].

Obviously, the same idea could be applied to any gradient dynamics. We could, for

example, minimize a different error function such as the Kullback-Leibler divergence of the

distribution which is to be learned and the distribution which is implemented by the vector

quantizer. This approach is not limited to supervised tasks. Unsupervised methods such as

the NG [9], which obey a gradient dynamics, could be improved by application of weighting

factors in order to obtain an adaptive metric. Furthermore, in unsupervised topographic

mapping models like the SOM or NG one can try to learn the relevance of input dimensions

subject to the requirement of topology preservation [33],[35].

3.4 Further Approaches

Beside the above introduced methods further approaches of neural inspired algorithms are

well known. In this context we have to refer to the family of Fuzzy-clustering algorithms as for

instance Fuzzy-SOM [36],[37],[38], Fuzzy-c-means [39],[40] or other (neural) vector quantizers

based on minimization of an energy function [41],[42],[20]. For an overview of resepective

approaches we refer to [43] (neural network approaches), [44],[45] (pattern classification).

4 Application in Remote Sensing Image Analysis

4.1 Low-Dimensional Data: LANDSAT TM Multi-spectral Im-

ages

4.1.1 Image Description

LANDSAT TM satellite-based sensors produce images of the Earth in 7 different spectral

bands. The ground resolution in meters is 30 30 for bands 1 5 and band 7. Band 6


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(thermal band) is often dropped from analyses because of the lower spatial resolution. The

LANDSAT TM bands were strategically determined for optimal detection and discrimination

of vegetation, water, rock formations and cultural features within the limits of broad band

multi-spectral imaging. The spectral information, associated with each pixel of a LANDSAT

scene is represented by a vector v V RDV with DV = 6. The aim of any classification

algorithm is to subdivide this data space into subsets of data points, with each subset

corresponding to specific surface covers such as forest, industrial region, etc. The feature

categories are specified by prototype data vectors (training spectra).

In the present contribution we consider a LANDSAT TM image from the Colorado area,

U.S.A.6 In addition we have a manually generated label map (ground truth image) for the

assessment of classification accuracy. All pixels of the Colorado image have associated ground

truth labels, categorized into 14 classes. The classes include several regions of different

vegetation and soil, and can be used for supervised learning schemes like the GRLVQ (see

Table 1).

4.1.2 Analyses of LANDSAT TM Imagery

A initial Grassberger-Procaccia analysis [46] yields DGP 3.1414 for the intrinsic dimen-

sionality (ID) of the Colorado image.

SOM analysis The GSOM generates a 12 7 3 lattice (Nmax = 256) in agreement

with the Grassberger-Procaccia analysis (DGP 3.1414), which corresponds to a P-value of

0.0095 indicating good topology preservation (Figure 3).

One way to visualize the GSOM clusters of LANDSAT TM data is to use a SOM di-

mension DA = 3 [47] and interpret the positions of the neurons r in the lattice A as vectors

c (r) = (r,g,b) in the RGB color space, where r,g,b are the intensities of the colors red, green

and blue, respectively, computed as cl (r) =(rl1)(nl1)

255, for l = 1, 2, 3 [47]. Such assignment of

colors to winner neurons immediately yields a pseudo-color cluster map of the original image

for visual interpretation. Since we are mapping the data clouds from a 6-dimensional input

space onto a three-dimensional color space dimensional conflicts may arise and the visual

interpretation may fail. However, in the case of topologically faithful mapping this color

representation, prepared using all six LANDSAT TM image bands, contains considerably

6 Thanks to M. Augusteijn (Univerity of Colorado) for providing this image.


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more information than a costumary color composite combining three TM bands (frequently

bands 2, 3, and 3). [47].

Application of GRLVQ We trained the GRLVQ with 42 codebook vectors (3 for eachclass) on 5% of the data set until convergence. The algorithm converged in less than 10

cycles for = 0.1 and 1 = 0.01. The GRLVQ produced a classification accuracy of about

90% on the training set as well as on the entire data set. The weighting factors resulting

from GRLVQ analysis provide a ranking of the data dimensions:

= (0.1, 0.17, 0.27, 0.21, 0.26, 0.0) (4.1)

Clearly, dimension 6 is ranked as least important with weighting factor close to 0. Dimension

1 is the second least important followed by dimensions 2 and 4. Dimensions 3 and 5 are

of similarly high importance according to this ranking. If we prune dimensions 6, 1, and

2, an accuracy of 84% can still be achieved. Pruning dimension 4 brings the classification

accuracy down to 50%, which is a very significant loss of information. This indicates that

the intrinsic data dimension may not be as low as 2. These classification results are shown

in Figure 4 with misclassified pixels colored black and classes keyed by the color wedge.

Use of the scaled metric (3.18) during training (which is equivalent to having a new

input space V) of a GSOM results in a two-dimensional lattice structure the size of which

is 11 10. This is in agreement with a corresponding Grassberger-Procaccia estimate of

the intrinsic data dimension as DGP 2.261 when this metric is applied. However, as we

saw above and in Figure 4, reducing the dimensionality of this Landsat TM image to 2 is

not as simple as discarding the four input image planes that received the lowest weightings

from the GRLVQ. Both the Grassberger-Procaccia estimate and the GSOM suggest the

intrinsic dimension of 2 for V whereas the dimension suggested by GRLVQ is at least 4.

Hence, the (scaled) data lie on a two-dimensional submanifold in V. The distribution of

weights for the two-dimensional lattice structure is visualized in Figure 5 for each original

(but scaled) input dimension. All input dimensions, except the fourth, seem to be correlated.

The fourth dimension shows a clearly different distribution which causes a two-dimensional

representation. The corresponding clustering, visualized in the two-dimensional (r,g, 0) color

space is depicted in Figure 4. Based on visual inspection this cluster structure compares

better with the reference classification in Figure 4, upper left, than the GSOM clustering


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in Figure 4. The scaling information provided to the GSOM by GRLVQ based on training

labels seems to have improved the quality of the clustering.

4.2 Hyperspectral Data: The Lunar Crater Volcanic Field

AVIRIS Image

4.2.1 Image Description

A Visible-Near Infrared (0.4 2.5 m), 224-band, 20 m/pixel AVIRIS image of the Lunar

Crater Volcanic Field (LCVF), Nevada, U.S.A., was analyzed in order to study SOM per-

formance for high-dimensional remote sensing spectral imagery. (AVIRIS is the Airborne

Visible-Near Infrared Imaging Spectrometer, developed at NASA/Jet Propulsion Labora-

tory. See http://makalu.jpl.nasa.gov for details on this sensor and on imaging spectroscopy.)

The LCVF is one of NASAs remote sensing test sites, where images are obtained regularly.

A great amount of accumulated ground truth from comprehensive field studies [48] and

research results from independent earlier work such as [49] provide a detailed basis for the

evaluation of the results presented here.

Figure 6 shows a natural color composite of the LCVF with labels marking the locations

of23 different surface cover types of interest. This 10 12km2 area contains, among other

materials, volcanic cinder cones (class A, reddest peaks) and weathered derivatives thereof

such as ferric oxide rich soils (L, M, W), basalt flows of various ages (F, G, I), a dry lake

divided into two halves of sandy (D) and clayey composition (E); a small rhyolitic outcrop

(B); and some vegetation at the lower left corner (J), and along washes (C). Alluvial material

(H), dry (N,O,P,U) and wet (Q,R,S,T) playa outwash with sediments of various clay contents

as well as other sediments (V) in depressions of the mountain slopes, and basalt cobble stones

strewn around the playa (K) form a challenging series of spectral signatures for pattern

recognition (see in [6]). A long, NW-SE trending scarp, straddled by the label G, borders

the vegetated area. Since this color composite only contains information from three selected

image bands (one Red, one Green, and one Blue), many of the cover type variations remain

undistinguished. They will become evident in the cluster and class maps below.

After atmospheric correction and removal of excessively noisy bands (saturated water

bands and overlapping detector channels), 194 image bands remained from the original 224.

These 194-dimensional spectra are the input patterns in the following analyses.


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The spectral dimensionality of hyperspectral images is not well understood and it is an

area of active research. While many believe that hyperspectral images are highly redundant

because of band correlations, others maintain an opposite view. Few investigations exist into

the intrinsic dimensionality (ID) of hyperspectral images. Linear methods such as PCA or

determination of mixture model endmembers [50] [51] usually yield 3 8 endmembers.

Optimally Topology Preserving Maps, a TRN approach [52], finds the spectral ID of the

LCVF AVIRIS image to be between 3 and 7 whereas the Grassberger-Procaccia estimate

[46] for the same image is DGP 3.06. These surprisingly low numbers, that increase with

improved sensor performance [53], result from using statistical thresholds for the determina-

tion of what is relevant, regardless of application dependent meaning of the data.

The number of relevant components increases dramatically when specific goals are con-

sidered such as what cover classes should be separated. With an associative neural network,

Pendock [54] extracted 20 linear mixing endmembers from a 50-band (2.0 2.5 m) seg-

ment of an AVIRIS image of Cuprite, Nevada (another well known remote sensing test site),

setting only a rather general surface texture criterium. Benediktsson et al. [55] performed

feature extraction on an AVIRIS geologic scene of Iceland, which resulted in 35 bands.

They used an ANN (the same network that performed the classification itself) for Decision

Boundary Feature Extraction (DBFE). The DBFE is claimed to preserve all features that

are necessary to achieve the same accuracy by using all the original data dimensions, by the

same classifier for predetermined classes. However, no comparison of classification accuracy

was made using the full spectral dimension to support the DBFE claim. In their study a

relatively low number of classes, 9, were of interest, and the goal was to find the number of

features that describe those classes. Separation of a higher number of classes may require

more features.

It is not clear how feature extraction should be done in order to preserve relevant in-

formation in hyperspectral images. Selection of 30 bands from our LCVF image by any of

several methods leads to a loss of a number of the originally determined 23 cover classes.

One example is shown in Figure 8. Wavelet compression studies on an earlier image of the

the same AVIRIS scene [56] conclude that various schemes and compression rates affect dif-

ferent spectral classes differently, and none was found overall better than another, within

25% 50% compressions (retaining 75% 50% of the wavelet coefficients). In a study on

simulated, 201-band spectral data, [57] show slight accuracy increase across classifications


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on 20-band, 40-band, and 60-band subsets. However, that study is based on only two vege-

tation classes, the feature extraction is a progressive hierarchical subsampling of the spectral

bands, and there is no comparison with using the full, 201-band case. Comparative studies

using full spectral resolution and many classes are lacking, in general, because few methods

can cope with such high-dimensional data technically, and the ones that are capable (such

as Minimum Distance, Parallel Piped) often perform too poorly to merit consideration.

Undesirable loss of relevant information can result using any of these feature extraction

approaches. In any case, finding an optimal feature extraction requires great preprocessing

efforts just to taylor the data to available tools. An alternative is to develop capabilities

to handle the full spectral information. Analysis of unreduced data is important for the

establishment of benchmarks, exploration and novelty detection; as well as to allow for

the distinction of significantly greater number of cover types than from traditional multi-

spectral imagery (such as LANDSAT TM), according to the purpose of modern imaging

spectrometers.

4.2.2 SOM Analyses of Hyperspectral Imagery

A systematic supervised classification study was conducted on the LCVF image

(Figure 6), to simultaneously assess loss of information due to reduction of spectral dimen-

sionality, and to compare performances of several traditional and an SOM-based hybrid

ANN classifier. The 23 geologically relevant classes indicated in Figure 6 represent a great

variety of surface covers in terms of spatial extent, the similarity of spectral signatures [6],

and the number of available training samples. The full study, complete with evaluations of

classification accuracies, is described in [58]. Average spectral shapes of these 23 classes are

also shown in [6].

Figure 7, top panel, shows the best classification, with 92% overall accuracy, produced

by an SOM-hybrid ANN using all 194 spectral bands for input. This ANN first learns in an

unsupervised mode, during which the input data are clustered in the hidden SOM layer. In

this version the SOM uses the conscience mechanism of DeSieno [18], which forces density

matching (i.e. a magnification factor of 1) as pointed out in Section 3.2 and illustrated

for this particular LCVF image and SOM mapping in [59]. After the convergence of the

SOM, the output layer is allowed to learn class labels via a Widrow-Hoff learning rule. The

preformed clusters in the SOM greatly aid in accurate and sensitive classification, by helping


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prevent the learning of inconsistent class labels. Detailed description of this classifier is

given in several previous scientific studies, which produced improved interpretation of high-

dimensional spectral data compared to earlier analyses [60] [61] [62]. Training samples

for the supervised classifications were selected based on field knowledge. The SOM hidden

layer was not evaluated and used for identification of spectral types (SOM clusters) prior

to training sample determination. Figure 7 reflects the geologists view of the desirable

segmentation.

In order to apply Maximum Likelihood and other covariance based classifiers, the number

of spectral channels needed to be reduced to 30, since the maximum number of training spec-

tra that could be identified for all classes was 31. Dimensionality reduction was performed

in several ways, including PCA, equidistant subsampling, and band selection by a domain

expert. Band selection by domain expert proved most favorable. Figure 7, bottom panel,

shows the Maximum Likelihood classification with 30 bands, which produced 51% accuracy.

A number of classes (notably the ones with subtle spectral differences, such as N, Q, R, S,

T, V, W) were entirely lost. Class K (basalt cobbles) disappeared from most of the edge of

the playa, and only traces of B (rhyolitic outcrop) remained. Class G and F were greatly

overestimated. Although the ANN classifier produced better results (not shown here) on the

same 30-band reduced data set than the Maximum Likelihood, a marked drop in accuracy

(to 75% from 92%) occurred compared to classification on the full data set. This emphasizes

that accurate mapping of interesting, spatially small geologic units is possible from full

hyperspectral information and with appropriate tools.

Discovery in Hyperspectral Images with a SOM The previous section demonstrated

the power of the SOM in helping discriminate among a large number of predetermined

surface cover classes with subtle differences in the spectral patterns, using the full spectral

resolution. It is even more interesting to examine the SOMs preformance for the detection

of clusters in high-dimensional data. Figure 8 displays a 40 40 SOM trained with the

the conscience algorithm [18]. The input data space was the entire 194-band LCVF image.

Groups of neurons, altogether 32, that were found to be sensitized to groups of similar

spectra in the 194-dimensional input data, are indicated by various colors. The boundaries

of these clusters were determined by a somewhat modified version of the U-matrix method

[63]. The density matching property of this variant of the SOM facilitates proper mapping


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of spatially small clusters and therefore increases the possibility of discovery. Examples of

discoveries are discussed below. Areas where no data points (spectra) were mapped are the

grey corners with uniformly high fences, and are relatively small. The black background in

the SOM lattice shows areas that have not been evaluated for cluster detection. The spatial

locations of the image pixels mapped onto the groups of neurons in Figure 8, are shown in the

same colors in Figure 9. Color coding for clusters that correspond to classes or subclasses of

those in Figure 7, top, is the same as in Figure 7, to show similarities. Colors for additional

groups were added.

The first observation is the striking correspondence between the supervised ANN class

map in Figure 7, top panel, and this clustering: the SOM detected all classes that were known

as meaningful geological units. The discovery of classes B (rhyolitic outcrop, white), F

(young basalt flows, dark grey and black, some shown in the black ovals), G (a different

basalt, exposed along the scarp, dark blue, one segment outlined in the white rectangle), K

(basalt cobbles, light blue, one segment shown in the black rectangle), and other spatially

small classes such as the series of playa deposits (N, O, P, Q, R, S, T) is significant. This

is the capability we need for sifting through high-volume, high-information-content data to

alert for interesting, novel, or hard-to-find units. The second observation is that the SOM

detected more, spatially coherent, clusters than the number of classes that we trained for in

Figure 7. The SOMs view of the data is more refined and more precise than that of the

geologists. For example, class A (in Figure 7) is split here into a red (peak of cinder cones)

and a dark orange (flanks of cinder cones) cluster, that make geologic sense. The maroon

cluster to the right of the red and dark orange clusters at the bottom of the SOM fills in some

areas that remained unclassified (bg) in the ANN class map, in Figure 7. An example is the

arcuate feature at the base of the cinder cone in the white oval, that apparently contains a

material different enough to merit a separate spectral cluster. This material fills other areas,

also unclassified in Figure 7, consistently at the foot of cinder cones (another example is seen

in the large black oval). Evaluation of further refinements are left to the reader. Evidence

that the SOM mapping in Figure 9 approximates an equiprobabilistic mapping (that the

magnification factor for the SOM in Figure 9 is close to 1), using DeSienos algorithm, is

presented in [59].


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GSOM Analysis of the LCVF Hyperspectral Image As mentioned above, earlier

investigations yielded an intrinsic spectral dimensionality of 3 7 for the LCVF data set

[52]. The Grassberger-Procaccia estimate [46] DGPA 3.06 corroborates the lower end of

the above range, suggesting that the data are highly correlated, and therefore a drastic

dimensionality reduction may be possible. However, a faithful topographic mapping is nec-

essary to preserve the information contained in the hyperspectral image. In addition to the

conscience algorithm explicit magnification control [19] according to (3.10) and the growing

SOM (GSOM) procedure, as extensions of the standard SOM, are suitable tools [64]. The

GSOM produced a lattice of dimensions 8 6 6 for the LCVF image, which represents a

radical dimension reduction, and it is in agreement with the Grassberger-Procaccia analysis

above. The resulting false color visualization of the spectral clusters is depicted in Figure 10.

It shows a segmentation similar to that in the supervised classification in Figure 7, top panel.

Closer examination reveals, however, that a number of the small classes with subtle spectral

differences from others, were lost (B, G, K, Q, R, S, T classes in Figure 7, top panel). In

addition, some confusion of classes can be observed. For example, the bright yellow cluster

appears at locations of known young basalt outcrops (class F, black, in Figure 7), and it also

appears at locations of alluvial deposits (class H, orange in Figure 7, top panel). This may

inspire further investigation into the use of magnification control, and perhaps the growth

criterium in the GSOM.

5 Conclusion

Self-Organizing Maps have been showing great promise for the analyses of remote sensing

spectral images. With recent advances in remote sensor technology, very high-dimensional

spectral data emerged and demand new and advanced approaches to cluster detection, vi-

sualization, and supervised classification. While standard SOMs produce good results, the

high dimensionality and large amount of hyperspectral data call for very careful evaluation

and control of the faithfulness of topological mapping performed by SOMs. Faithful topolog-

ical mapping is required in order to avoid false interpretations of cluster maps created by an

SOM. We summarized several new approaches developed in the past few years. Extensions to

the standard Kohonen SOM, the Growing Self-Organizing Map, magnification control, and

Generalized Relevance Learning Vector Quantization were discussed along with the modified


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topographic product P. These ensure topology preservation through mathematical consid-

erations. Their performance, and relationship to a former powerful SOM extension, the

DeSieno conscience mechanism was discussed in the framework of case studies for both low-

dimensional traditional multi-spectral, and very high-dimensional (hyperspectral) imagery.

The Grassberger-Procaccia analysis served for an independent estimate of the determina-

tion of intrinsic dimensionality to benchmark ID estimation by GSOM and GRLVQ. While

we show some excellent data clustering and classification, there remains certain discrepancy

between theoretical considerations and application results, notably with regard to intrinsic

dimensionality measures and the consequences of dimensionality reduction to classification

accuracy. This will be targeted in future work. Finally, since it is outside the scope of this

contribution, we want to point out that full scale investigations such the LCVF study also

have to make heavy use of advanced image processing tools and user interfaces, to handle

great volumes of data efficiently, and for effective graphics/visualization. References to such

tools are made in the cited literature on data analyses.

6 Acknowledgements

E.M. has been supported by the Applied Information Systems Research Program of NASA,

Office of Space Science, NAG59045 and NAG5-10432. Contributions by Dr. William H.

Farrand, providing field knowledge and data for the evaluation of the LCVF results, are

gratefully acknowledged. Khoros (Khoral Research, Inc.) and NeuralWorks Professional

(NeuralWare) packages were utilized in research software development.

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Tables


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30

class label ground cover type

a scotch pine

b Douglas fir

c pine/fir

d mixed pine forrest

e supple/prickle pine

f aspen / mixed pine forrest

g without vegetation

h aspen

i water

j moist meadow

k bush land

l grass / pastureland

m dry meadow

n alpine vegetation

o misclassification

Table 1: Table of labels and classes occuring in the LANDSAT TM Colorado image.


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Figures


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Figure 1: Comparison of the spectral resolution of the AVIRIS hyperspectral imager and the

LANDSAT TM imager in the visible and near-infrared spectral range. AVIRIS representsa quasi-continuous spectral sampling whereas LANDSAT TM has coarse spectral resolution

with large gaps between the band passes. Additionally, spectral reflectance charateristics of

common earth surface materials is shown: 1 - water, 2 - vegetation, 3 - soil (from [2]).


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Figure 2: Left: The concept of hyperspectral imaging. Figure from [4]. Right: The spectral

signature of the mineral alunite as seen through the 6 broad bands of Landsat TM, as seen by

the moderate spectral resolution sensor MODIS (20 bands in this region), and as measured

in laboratory. Figure from [5]. Hyperspectral sensors such as AVIRIS of NASA/JPL [3]

produce spectral details comparable to laboratory measurements.


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Figure 3: Left: The reference classification for the Landsat TM Colorado image, provided

by M. Augusteijn. Classes are keyed as shown on the color wedge on the left, and the

corresponding surface units are described in Table 1. Right: Cluster map of the Colorado

image derived from all six bands by the 12 7 3 GSOM-solution. Note that, due to the

visualization scheme, which uses a continuous scale of RGB colors described in the text,

this map does not show hard cluster boundaries. More similar colors indicate more similar

spectral classes.


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Figure 4: GRLVQ results for the Landsat TM Colorado image. Upper left: GRLVQ

without pruning. Upper right: GRLVQ with pruning of dimensions 1, 2, and 6. Lower

left: GRLVQ with pruning dimensions 1, 2, 6, and 4. Lower right: Clustering of the

Colorado image using GSOM with GRLVQ scaling. For the three supervised classifications,

the same color wedge and class labels apply as in Figure 3, left.


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Figure 5: The distribution of weight values in the two-dimensional 1110 GSOM lattice for

the non-vanishing input dimensions 15 as a result of GRLVQ-scaling of the Colorado image

during GSOM training. Dimension 4 has significantly different weight distribution from all

others, while in the rest of the dimensions the variations are less dramatic. (Please note that

the scaling is different for each dimension.) For visualization the SOM-Toolbox provided by

the Neural Network Group at Helsinki University of Technology was used.


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Figure 6: The Lunar Crater Volcanic Field. RGB natural color composite from an AVIRIS,

1994 image. The full hyperspectral image comprises 224 image bands over the 0.4 2.5

m wavelength range, 512 614 pixels, altogether 140 Mbytes of data. Labels indicate 23

different cover types described in the text. The ground resolution is 20 m/pixel.


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Figure 7: Top: SOM-hybrid supervised ANN classification of the LCVF scene, using 194

image bands. The overall classification accuracy is 92% [58]. Bottom: Maximum Likelihood

classification of the LCVF scene. 30, strategically selected bands were used due to the limited

number of training samples for a number of classes. Considerable loss of class distinction

occurred compared to the ANN classifi

cation, resulting in an overall accuracy of 51%. bgstands for background (unclassified pixels).


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Figure 8: Clusters identified in a 4040 SOM. The SOM was trained on the entire 194-band

LCVF image, using the DeSieno algorithm [18].


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Figure 9: The clusters from Figure 8 remapped to the original spatial image, to show where

the different spectral types originated from. The relatively large, light grey areas correspond

to the black, unevaluated parts of the SOM in Figure 8. Ovals and rectangles highlight

examples of small classes with subtle spectral differences, discussed in the text. 32 clusters

were detected, and found geologically meaningful, adding to the geologists view reflected in

Figure 7, top panel.


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Figure 10: GSOM-generated cluster map of the same 194 band hyperspectral image of the

Lunar Crater Volcanic Field, Nevada, USA, as in Figure 7. It shows groups similar to those

in the supervised classification map in Figure 7, top panel. Subtle spectral units such as B,

G, K, Q, R, S, T were not separated, however, and for example, the yellow unit appears at

both the locations of known young basalt deposits (class F, black, in Figure 7, top panel)

and at locations of alluvial deposits (class H, orange, in Figure 7).

thomas villmann, erzsébet merényi and barbara hammer- neural maps in remote sensing image analysis

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